A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Waiting time probability question

I want to solve the following problem: A dentist works 4 hours a day. Patients arrive on the average of 1 per 20 minutes and one patient spends on average 15 minutes with the dentist. Both time ...
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1answer
21 views

Poisson process and moment generating function

If we have a Poisson process $ \lbrace N(t), t > 0 \rbrace $ with rate $\lambda > 0 $ and if we have a random variable $S$ having a uniform distribution on the interval $[0,2]$. I was wondering ...
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11 views

Construct SDE with two uncorrelated Brownian motions

Using $Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$ as a model to construct a process where X1 and X2 are brownian motions with drifts and brownian increments $dX_1(t)= \mu_1dt + \sigma_1dW_1(t)$ ...
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1answer
13 views

Covariance function meaning

I have this sentence in a report but I don't quiet know what it means. I am familier with covariance and covariance matrices but not with covariance functions. $f(t)$ is a continuous-time ...
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7 views

Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
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7 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
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14 views

Branching processes, extinction probability

Why do we assume that a branching process starts with one ancestor. What happens to extinction probability if we have more than one ancestor in generation Y(0)?
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1answer
10 views

Ornstein-Uhlenbeck process written explicitly

I need to show that the Ornstein-Uhlenbeck process, $$ dX_t = -\theta X_tdt + dB(t) $$ Where $X_0=0$, $B(t)$ is Brownian motion and $\theta>0$ can be written explicitly as: $$ X_t=B(t) - \theta ...
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1answer
7 views

First property of discrete time homogenous markov chain

I'm trying to understand the properties of a DTHMC. I am having trouble understanding with the first one. My textbook says - "$X_t$ takes values in $X$ for all $t$ (i.e. $X_t$ is a random variable ...
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1answer
14 views

Showing a process is a martingale from Ito's lemma

Suppose that we have the process $t-W^2(t)$ where $W(t)$ is a Brownian motion with filtration $\mathcal{F}_t$. It is easy to show that this is a martingale by computing ...
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1answer
14 views

Solving the SDE $dX_t=bdt+cX_t dW_t$

I want to solve the SDE $dX_t=bdt+cX_t dW_t$, $X_0=0$ for $b,c\in\mathbb R$. I start by rewriting this as $$dX_t=(\mu_1+\mu_2 X_t )dt+(\sigma_1+\sigma_2 X_t )dW_t$$ where $\mu_1=b, \mu_2=0, ...
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21 views

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
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31 views

Poisson process: Has my book used a necessary condition, when it should have used a sufficient condition?

My book says that if we know that if we are viewing a poisson process with length $t$, and know that $n$ events happened in that interval, than the time that any of those events happened is uniformly ...
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14 views

Variance of exit time for simple symmetric random walk

For a simple symmetric random walk starting at 0 (that is, a Markov chain on the integers starting at 0 with equal probabilities of going to the left and right at each step), I want to compute the ...
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1answer
11 views

M/M/1 queue with probability of new client leaving

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
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1answer
21 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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20 views

Showing the square of a Markov process is or isn't Markov

Hi I am trying to show that if $X_n$ is a markov process, whether or not $X_n^2$ is a markov process. $X_n$ is a markov process if $P\{X_k = a_k|X_{k-1} = a_{k-1}, X_{k-2} = a_{k-2}, ..., X_k = a_1 ...
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1answer
22 views

How are the waiting times distributed, poisson process.

I am wondering how the waiting times are distributed for the poisson process, conditioned on a number of events by time t. Look at this theorem: Here, the S's are the sum of the waiting time to ...
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1answer
30 views

Solution to a stochastic differential equation

I could really do with some help on this question, have no idea where to start. Any advice would be much appreciated, thank u in advance. I am given $$\begin{align}dx(t)&=(1+x(t))dt + x(t) ...
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1answer
29 views

Find the distribution of the maximum of a Wiener Process with negative drift

So.. what I have now is Let $M=max\{W_t; 0\leq t <\infty\}$ since $W_0=0$, $M\geq 0$ with probability 1. So, $P(M>x)=P(T_x<\infty)$ where $T_x$ is the stopping time, so we now use the ...
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38 views

Markovian birth-death process [on hold]

A linear Markovian death process, initialized at five members, experiences an average daily death rate $u=0.1$. Determine the probability of having fewer than three members in the population after a ...
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1answer
31 views

Gaussian vectors and covariance matrix.

The following is a part of a question I was given in stochastic processes course. It goes like this - I am given a series of gaussian iid random variables $\{V_i\}_{i=1}^N$ , the variable $X_0 \sim ...
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19 views

Distribution and Laplace transform

I'm having trouble understanding this problem from Resnick's Adventures in Stochastic Processes: The problem says: Suppose $F$ is a distribution of a positive random variable and $p_k \geq 0, ...
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1answer
25 views

Variance of sum of Brownian Motions

Let $t_i=\frac{T\cdot i}{n}$ for $T>0$, $i=1,...,n$ and let $(W_t)_{0\le t\le T}$ be a standard Brownian motion. Now I want to evaluate $$\text{var}\left(\sum_{i=1}^n W_{t_i}\right) = \mathbb ...
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28 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
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8 views

Karhunen– Loeve expansion analytical solution confusion - which is correct?

Looking at the analytical solution to the following: $cov \:u =\lambda\: u$ With this kernel: $cov(x',x) = \sigma^2 e^{-{1\over b}|x'-x|}$ O. P. Le Maître, Omar M. Knio (Spectral Methods for ...
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1answer
27 views

Probability of having 2 consequent 1's in a random finite sequence

Fix natural numbers $t,n,k$. Consider the following stochastic process for generation of finite sequences of elements from $\{1,\ldots,n\}$: $\sigma_0$ is the empty sequence. Suppose we have ...
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2answers
34 views

$1/(1+X_n)$ bounded in probability

I am trying to prove that if $X_n\rightarrow 0$ in probability, then $1/(1+X_n)$ is bounded in probability. My attempt is: $$P(\frac{1}{1+X_n}<\frac{1}{1-\epsilon})=P(|1+X_n|>1-\epsilon)\\ \geq ...
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12 views

Convergence of Quantiles moments.

QUESTION: Let $F$ be an absolutely continuous distribution function with density f, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
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1answer
52 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
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1answer
39 views

Deriving the PDE for basket option

The payoff for basket option is max($w_1S_1+w_2S_2 -k,0)$. Using Ito's formula, I need to derive the PDE, where $dS_1 = rS_1dt + \sigma_1 S_1dW_1$ $dS_2 = rS_2dt + \sigma_2 S_2dW_2$ I need some ...
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20 views

Error in thinking: Poisson Process is a Markov Process

I am a bit confused on proving the Markov property for Poisson processes. That is, we want to prove, if $X = (X_t: t \in \mathbb{R})$ is a Poisson process with rate $\lambda$: $P(X_{t_n} = a_n | ...
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10 views

A Question on the independence of the sample mean and sample variance

The aim of the following question is to show the given random variable follows a student T distribution. Although it seems quite straightforward at the first sight, I am quite confused about the ...
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30 views

continuous time markov process - first passage time

Let $(X_t)_{t\ge0}$ is a continuous time-homogeneous Markov diffusion process such that $X_0=y$. Let $$p(x,t|y)=d\Pr(X_t\le x|X_0=y)/dx$$ be the respective transition probability density. Let ...
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17 views

Sufficient conditions for Uniform Law of Large Numbers

I would need a Uniform Law of Large numbers for $f_T(\theta)$ over $\Theta$ when $f$ is the indicator function and, thus, not continuous over $\Theta$. Do you know about any sufficient conditions?
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1answer
29 views

relation between multivariate probability generating function and univariate ones

Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I ...
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8 views

Is reflected levy process a feller process?

In some literature , there is a concept similar to reflected Brownian process. Assume that $L_{t}$ is a levy process (may be we can assume it's not a Poisson process) then reflected Levy process ...
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25 views

What is Poisson Point Process?

"Points $\{A_j\}_{j\in\Phi(\lambda)}$ are assumed to be distributed according to a homogeneous PPP with intensity $\lambda$, denoted $\Phi(\lambda)=\{X_j\}$, where $X_j$ is the location of the $j$th ...
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21 views

Is convergence in probability to a uniformly continuous function a sufficient condition for stochastic equicontinuity?

Suppose that a random function $g_T(\theta)$ converges in probability to a function $g(\theta)$ uniformly continuous over $\Theta$ as $T\rightarrow \infty$ $\forall \theta \in \Theta$. Is this ...
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65 views

Stopping time and filtration

My question is as follow: Let $(\Omega,\cal{F}_\infty,\{\cal{F}_t\},\mathbb{P})$ be the filtred probability space. Further, denote $\cal{F}^*_t$ as the usual augmented filtration. Now, given a ...
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1answer
30 views

multivariate probability generating function

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose ...
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1answer
23 views

Question about limit of Stochastic Process

Given $\mu_t$ continuous stochastic process that satisfies $\int_0^t \mu_s^2\;ds<\infty$. Define $X_t\equiv \int_0^t \mu_s\;ds$. Let $|\cdot|$ denote floor function. Then where does ...
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1answer
25 views

Mean time spent in transient states/Markov chain

I dont get this in my book: For transient states $i$ and $j$ , let $s_{ij}$ denote the expected number of time periods that the markov chain is in state $j$ , given that it starts in state $i$. Let ...
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linear system output when input is a Gaussian process?

Rectently, I read a technical book that says:" the linear transform of a Guassian process is also a Guassian process. i.e. for continuous time case: $$ x(t)*h(t)=y(t)$$ the input $x(t)$ is a ...
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1answer
22 views

$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
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1answer
31 views

Coupled stochastic differential equations?

I'm a physics student working on a quantum information project (so please be gentle with me). My work involves stochastic processes and I'm new to the topic, so I'm asking some help about a system of ...
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34 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
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2answers
41 views

Is this true about Brownian Motion?

I have the following in my notes and I'm not sure if it's true or not. Any help would be highly appreciated. If $\{W_t\}_{t\geq0}$ is a standard Brownian motion stochastic process, $\Delta>0$ and ...
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22 views

Why is chemotaxis considered an emergent behavior?

this is an applied math question. I could have posted this under a biological stackexchange, but the idea of emergent behavior or emergent properties of a system seems more appropriate to an applied ...
2
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32 views

Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...